Would you consider moving further out?? There are some good train links to Kings Cross in Herts, schools tend to be better overall (higher averages) and property is cheaper. But I know some people find it hard to leave London!

With either palmers green or enfield you need to be looking at local schools - check out their ofsted reports, websites etc. Both areas come under Enfield council, and on their website there's a school application booklet that tells you the maximum 'catchment' distance from each school (not really a catchment area, but gives you an idea of how far you can live from the school and stand a chance of your child getting in) in the past years.